Let \(\theta =\lbrace  \theta _Q, \theta _D \rbrace \)  be the parameters of the neural dense retrieval model of Equation REF . \(\theta \)  is learned to identify the most effective query and document representations for document retrieval. Consider a query \(Q\)  and two candidate documents \(D^+\)  and \(D^-\) , where \(D^+\)  is more relevant than \(D^1\)  to \(Q\) . Let \(\text{Rel}(Q,D; \theta )\) , as defined in Equation REF , be the relevance score of \(D\)  given \(Q\) . We want to maximize \(\Delta = \text{Rel}(Q,D^+; \theta ) - \text{Rel}(Q,D^-; \theta )\) . We do so by minimizing a smooth pair-wise rank loss [1]} defined as
\(\mathcal {L}(\Delta ; \theta ) &= \log (1 + \exp (-\gamma \Delta )), \) 
